Columbia UniversityDepartment of Physics

QUALIFYING EXAMINATION

Monday, January 9, 201711:00AM to 1:00PM

Classical PhysicsSection 1. Classical Mechanics

Two hours are permitted for the completion of this section of the examination. Choose4 problems out of the 5 included in this section. (You will not earn extra credit by doing anadditional problem). Apportion your time carefully.

Use separate answer booklet(s) for each question. Clearly mark on the answer booklet(s)which question you are answering (e.g., Section 1 (Classical Mechanics), Question 2, etc.).

Do NOT write your name on your answer booklets. Instead, clearly indicate your ExamLetter Code.

You may refer to the single handwritten note sheet on 812” × 11” paper (double-sided) you

have prepared on Classical Physics. The note sheet cannot leave the exam room once theexam has begun. This note sheet must be handed in at the end of today’s exam. Pleaseinclude your Exam Letter Code on your note sheet. No other extraneous papers or booksare permitted.

Simple calculators are permitted. However, the use of calculators for storing and/or recov-ering formulae or constants is NOT permitted.

Questions should be directed to the proctor.

Good Luck!

Section 1 Page 1 of 6

1. A particle of mass m is constrained to slide without friction along a straight rod, as inthe figure below. The rod is rotating in a vertical plane about one end with a constantangular velocity ω, in the presence of a uniform gravitational field g. The rod is horizontalat t = 0.

(a) Find the Lagrangian and the differential equations of motion for the mass m.

(b) Solve the differential equations assuming that at t = 0 the mass is at rest a distanced from the fixed end of the rod.

(c) Show that there is a certain critical angular frequency ωc such that, if ω < ωc,the mass will eventually hit the fixed end of the rod and, if ω > ωc, the mass willeventually fly off the rotating end of the rod. Find ωc.

g

Fixed End

mω

Section 1 Page 2 of 6

2. In Newtonian physics, the circular orbit of two gravitationally bound mass M objectsis a stable configuration. General relativity teaches us this is an approximation, as thesystem must lose energy due to gravitational radiation.

Consider the case of two mass M black holes in a circular orbit of very large radiusR. Using only Newtonian physics, estimate the total energy lost due to gravitationalradiation before the black holes merge. You may assume the black holes merge whenthey first touch, and that the radius of the black hole is given by the Newtonian conditionfor the escape velocity to be the speed of light.

Section 1 Page 3 of 6

3. A simple pendulum of mass m and length l is connected at its point of support to ablock of mass M . The block is free to slide along a frictionless horizontal track.

(a) Find the Lagrangian for this system (assuming it is placed in a uniform gravitationalfield). Find Lagrange’s equations.

(b) What are the constants of motion?

(c) What are the eigenfrequencies associated with small oscillations about the equilib-rium position of this system? Describe the motion of the system associated witheach frequency.

Section 1 Page 4 of 6

4. A symmetrical top with I1 = I2 6= I3 is initially placed with its center of mass at theorigin of a fixed inertial coordinate system XY Z and with the z body axis (axis 3) alongthe Z axis as indicated in the figure. At t = 0 the top spins about its z body axis withangular frequency Ω0. Between t = 0 and t = t0, the top simply spins about the z axiswith its center of mass fixed. At t0, two equal and opposite forces of magnitude F areexerted in the +X and −X direction and at Z = +l and −l respectively. The forces arevery large but act for a short time ∆t with F∆t of finite size. Give the motion of thetop subsequent to the application of the forces F and −F in the absence of gravity.

Y

X

ZΩ0

0 < t < t0

Y

X

Z

F

Fl

l

t = t0

Section 1 Page 5 of 6

5. A cylinder with mass m and radius r is suspended from the ceiling as shown in Fig. 1.On one side, a spring of stiffness k is inserted. There is sufficient friction between therope and the cylinder to prevent sliding of the cylinder. Find the oscillation frequencyof the suspended cylinder.

m

r

k

Figure 1: A suspended cylinder.

Section 1 Page 6 of 6

1. Classical Mechanics: A particle of mass m is constrained to slide without friction along a

straight rod, as in the figure below. The rod is rotating in a vertical plane about one end with a

constant angular velocity !, in the presence of a uniform gravitational field g. The rod is horizontal

at t = 0.

a. (6 pts) Find the Lagrangian and the di↵erential equations of motion for this system.

b. (5 pts) Solve the di↵erential equations assuming that at t = 0 the mass is at rest a distance

d from the fixed end of the rod.

c. (4 pts) Show that there is a certain critical angular frequency !c such that, if ! < !c, the

mass will eventually hit the fixed end of the rod and, if ! > !c, the mass will eventually fly

o↵ the rotating end of the rod. Find !c.

2. Relativity: In Newtonian mechanics, power can be found from P =

dEdt =

~F · ~v, where E is

the total energy of a particle moving with velocity ~v and acted on by a net force

~F . Show that

this relation is also valid in relativistic mechanics (assume that Newton’s second law is valid under

special relativity).

1

Humensky Section 1-1

Quals ‐ Mechanics SolutionSaturday, November 19, 2016 3:51 PM

November 2016 Page 1

Humensky Section 1-1

November 2016 Page 2

HumenskySection 1-1

W.A. Zajc

1 Problem: Black Hole Merger

In Newtonian physics, the circular orbit of two gravitationally bound mass M objects is a stable configuration.General relativity teaches us this is an approximation, as the system must lose energy due to gravitationalradiation.

Consider the case of two mass M black holes in a circular orbit of very large radius R. Using only Newtonianphysics, estimate the total energy lost due to gravitational radiation before the black holes merge. You mayassume the black holes merge when they first touch, and that the radius of the black hole is given by theNewtonian condition for the escape velocity to be the speed of light.

1

Zajc Section 1-2

2 Solution: Black Hole Merger

Re-derive virial theorem averages to get factors of 2 right:

GM2

(2r)2=

Mv2

r⇒ KETot = Mv2 =

GM2

4r= −1

2VPot

⇒ ETot = KETot + VPot =1

2VPot = −

1

2

GM2

2rX

Schwarzschild radius given by

GMs

Rs=

1

2mc2 Rs =

2GM

c2,

which happens to be exactly correct for static black hole.

“very large radius R” ⇒ ETot ' 0 initially.

The final total energy is for r = Rs ⇒

EfinalTot = −GM2

4Rs= − GM2

4 · 2GM/c2= −1

8Mc2 ,

which is not too far off the LIGO observed value of (3.0 ± 0.5)M for the merger of a 29M and 36M blackholes.

This problem could be extended to calculate the power radiated, but this would require the understanding thatnearly all the radiation takes place in the last couple of orbits.

2

Zajc Section 1-2

2017 Quals Problem: Classical Mechanics

A simple pendulum of mass m and length l is connected at its point of support to a blockof mass M . The block is free to slide along a frictionless horizontal track.

a. Find the Lagrangian for this system (assuming it is placed in a uniform gravitationalfield). Find Lagrange’s equations.b. What are the constants of motion?c. What are the eigenfrequencies associated with small oscillations about the equilibriumposition of this system? Describe the motion of the system associated with each frequency.

Rosen Section 1-3

2017 Quals Problem: Classical MechanicsSolution

a.Kinetic energy: T = 1

2M(x2M + y2M) + 1

2m(x2m + y2m),

Potential energy: V = mgym,where xM = x, yM = 0, xm = x+ l sin θ and ym = −l cos θ.

Lagrangian:L = T − V = 1

2(M +m)x2 + 1

2ml2θ2 +ml cos θ θx+mgl cos θ

Lagrange’s equations:For x: (M +m)x+ml cos θ θ −ml sin θ θ2 = 0For θ: ml2θ +ml cos θ x−ml sin θ xθ +mgl sin θ = 0

b.Constants of motion: d

dt

(∂L∂x

)= 0 so ∂L

∂x= (M +m)x+ml cos θ θ = const

Also: E = T + V = const

c.Small angle approx:

T =

(M +m mlml ml2

), V =

(0 00 mgl

)So:

det

(−ω2(M +m) −ω2ml

−ω2ml +mgl − ω2ml2

)= 0

Eigenfrequencies:ω2 = 0: corresponds to the block and the pendulum moving together. I.e., x changes butnot θ.ω2 = g

lM+mM

: corresponds to oscillations of the pendulum. In the limit M m, the blockdoesn’t move.

Rosen Section 1-3

MuellerSection 1-4

Zelevinsky Section 1-5